a First we want to prove that lines a and c are parallel, knowing the fact that m∠ 1+m∠ 2=180. Let's begin by highlighting ∠ 1, ∠ 2, and ∠ 3 in the given diagram.
We can see that ∠ 2 and ∠ 3 form a linear pair, so they are supplementary. This means that the sum of the measures of these angles is 180.
m∠ 2 + m∠ 3 = 180
Since m∠ 1 + m∠ 2 = 180, we can equate these two equations to get an important result.
b Now we want to prove that lines t and c are perpendicular, knowing that lines a and c are parallel, and m∠ 1+m∠ 3 =180. Since a∥ c, and ∠ 1 and ∠ 3 are corresponding angles, by the Corresponding Angles Theorem they are congruent. Let's highlight this in the diagram.
We are told that m∠ 1 + m∠ 3 = 180. Now, we can substitute m∠ 1 = m∠ 3 into this last equation. This will let us find the measure of ∠ 3.
